EXHAUSTION, METHOD OF

in mathematics, technique invented by the classical Greeks to prove propositions regarding the areas and volumes of geometric figures. Although it was a forerunner of the integral calculus, the method of exhaustion used neither limits nor arguments about infinitesimal quantities. It was instead a strictly logical procedure, based upon the axiom that a given quantity can be made smaller than another given quantity by successively halving it (a finite number of times). From this axiom it can be shown, for example, that the area of a circle is proportional to the square of its radius. The term method of exhaustion was coined in Europe after the Renaissance and applied to the rigorous Greek procedures as well as to contemporary proofs of area formulas by exhausting the area of figures with successive polygonal approximations.